aa r X i v : . [ m a t h . R T ] A p r THE PBW THEOREM FOR THE AFFINE YANGIANS
YAPING YANG AND GUFANG ZHAO
Abstract.
We prove that the Yangian associated to an untwisted symmetric affine Kac-MoodyLie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed in[YZ14] as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtainthe Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independentproof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt [GRW18].
Introduction
For Yangians of a semi-simple Lie algebra, the PBW theorem is well-known. It has been con-jectural that the PBW theorem holds for a much larger class of Yangians including the affineYangians.On the other hand, the PBW theorem for the (undeformed) current algebra of an affine Liealgebra has been proved by Enriquez [E03]. In the proof of loc. cit. , a shuffle algebra descriptionand a duality of the current algebra play the key roles.In the framework of cohomological Hall algebras, in [YZ14] the authors of the present paper gavea conjectural shuffle algebra description of the Yangian of any symmetric Kac-Moody Lie algebra,and constructed a surjective map from the Yangian to the shuffle algebra. In the present paper,we combine this result and the earlier results of Enriquez [E03] to prove the PBW theorem for theYangian of any untwisted symmetric affine Lie algebra.For Y ~ ( b sl n ), this theorem has been proved by N. Guay in [G07]. During the preparation of thepresent paper, Guay-Regelskis-Wendlandt communicated to the authors an independent proof ofthe PBW theorem for Y ~ ( g KM ) using the vertex operator representations of the affine Yangians[GRW18].This paper is organized as follows. We start by recollecting relevant results from [YZ16]. Thestatements of the main theorems are in § § § § The affine Yangian and the shuffle algebra
The Yangian.
Let A = ( c ij ) i,j ∈ I be a symmetrizable generalized Cartan matrix. Thus, c ii = 2for any i ∈ I , c ij ∈ Z ≤ for any i = j ∈ I . Let g KM be the Kac-Moody Lie algebra associated to A .Denote by h the Cartan subalgebra of g KM , and { α i } i ∈ I a set of simple roots of g KM . The algebra Date : April 13, 2018.
Key words and phrases.
Yangian, PBW theorem, shuffle algebra, Drinfeld double.We are grateful to Hiraku Nakajima for the helpful suggestions, and to Nicolas Guay and Curtis Wendlandt forthe useful discussions and communications. The research of G.Z. at IST Austria, Hausel group, is supported by theAdvanced Grant Arithmetic and Physics of Higgs moduli spaces No. 320593 of the European Research Council. Y ~ ( g KM ) is denoted by Y ~ ( g ′ KM ) in [GNW17] which is the version of the Yangian of g KM withoutthe degree operator. Definition 1.1.
The Yangian Y ~ ( g KM ), is an associative algebra over C [ ~ ], generated by the vari-ables x ± k,r , ξ k,r , ( k ∈ I, r ∈ N ) , subject to relations described below. Take the generating series ξ k ( u ) , x ± k ( u ) ∈ Y ~ ( g Q )[[ u − ]] by ξ k ( u ) = 1 + ~ X r ≥ ξ k,r u − r − and x ± k ( u ) = ~ X r ≥ x ± k,r u − r − . The following is a complete set of relations defining Y ~ ( g KM ) (see, e.g., [GTL16, Proposition 2.3]): (Y1): For any i, j ∈ I , and h, h ′ ∈ h [ ξ i ( u ) , ξ i ( v )] = 0 , [ ξ i ( u ) , h ] = 0 , [ h, h ′ ] = 0 (Y2): For any i ∈ I , and h ∈ h ,[ h, x ± i ( u )] = ± α i ( h ) x ± i ( u ) (Y3) : For any i, j ∈ I , and a = ~ c ij ( u − v ∓ a ) ξ i ( u ) x ± j ( v ) = ( u − v ± a ) x ± j ( v ) ξ i ( v ) ∓ ax ± j ( u ∓ a ) ξ i ( u ) (Y4): For any i, j ∈ I , and a = ~ c ij ( u − v ∓ a ) x ± i ( u ) x ± j ( v ) = ( u − v ± a ) x ± j ( v ) x ± i ( u ) + ~ (cid:16) [ x ± i, , x ± j ( v )] − [ x ± i ( u ) , x ± j, ] (cid:17) (Y5) : For any i, j ∈ I ,( u − v )[ x + i ( u ) , x − j ( v )] = − δ ij ~ ( ξ i ( u ) − ξ i ( v )) (Y6) : For any i = j ∈ I , X σ ∈ S − cij [ x ± i ( u σ (1) ) , [ x ± i ( u σ (2) ) , [ · · · , [ x ± i ( u σ (1 − c ij ) ) , x ± j ( v )] · · · ]]] = 0 . Let Y + ~ ( g KM ) (resp. Y − ~ ( g KM )) be the algebra generated by the elements x + k,r , (resp. x − k,r ) for k ∈ I, r ∈ N , subject to the relations (Y4) and (Y6) with the + sign (resp. with the − sign).Let Y ≥ ~ ( g KM ) be the algebra generated by the elements x + k,r , h k,r , for k ∈ I, r ∈ N , subject to therelations (Y1), (Y2), (Y3) , (Y4), and (Y6) with the + sign.Let Y = Y ~ ( g KM ) be the polynomial algebra generated by ξ k,r , ( k ∈ I, r ∈ N ). We note thatfrom the relation (Y3) it is clear that there is a surjective map Y + ~ ( g KM ) ⋊ Y → Y ≥ ~ ( g KM )(see, e.g., [GTL10, Lemma 2.9]). Each of these algebras has an obvious algebra homomorphism to Y ~ ( g KM ). However, it is a priori not clear that Y + ~ ( g KM ) or Y ≥ ~ ( g KM ) are subalgebras of Y ~ ( g KM ).These facts only follow from § The shuffle algebra.
The shuffle algebra associated to any formal group law is introduced in[YZ14, YZ16]. It is an algebraic description of the cohomological Hall algebra (COHA) associatedto a preprojective algebra. The shuffle formula in [YZ18] describes the action of the COHA on theNakajima quiver varieties algebraically. In this section, we focus on the case when the formal grouplaw is additive, and recall the definition of the shuffle algebra introduced in [YZ14].
HE PBW THEOREM FOR THE AFFINE YANGIANS 3
The algebra structure.
Now we assume the Cartan matrix of g KM is symmetric. Let Q =( I, H ) be the associated Dynkin quiver of the symmetric Kac-Moody Lie algebra g KM with vertexset I and arrow set H . For each arrow h ∈ H , we denote by in( h ) (resp. out( h )) the incoming(resp. outgoing) vertex of h . Thus, the vertex set I is the index set of the simple roots of g KM , andthe arrow set H encodes the Cartan matrix of g KM .Let SH be an N I -graded C [ ~ ]-algebra. As a C [ ~ ]-module, we haveSH = M ~v ∈ N I SH ~v , where SH ~v := C [ ~ ] ⊗ C [ λ is ] S ~v i ∈ I,s =1 ,...,v i , here S ~v = Q i ∈ I S v i is the product of symmetric groups, and S ~v naturally acts on the variables { λ is } i ∈ I,s =1 ,...,v i by permutation. For any ~v and ~v ∈ N I , we consider SH ~v ⊗ C [ ~ ] SH ~v as a subalge-bra of C [ ~ ][ λ is ] { i ∈ I,s =1 ,..., ( v i + v i ) } by sending λ ′ is ∈ SH ~v to λ is , and λ ′′ is ∈ SH ~v to λ is + v i .The factor fac ~v , ~v is defined as follows. For any pair of vertices i and j with arrows h , . . . , h a from i to j , we associated to each arrow h p the pairs of integers m h p = a +2 − p and m h ∗ p = − a +2 p .Then,fac ~v , ~v := Y i ∈ I v i Y s =1 v i Y t =1 λ ′ is − λ ′′ it + ~ λ ′ is − λ ′′ it ·· Y h ∈ H (cid:16) v out( h )1 Y s =1 v in( h )2 Y t =1 ( λ ′′ in( h ) t − λ ′ out( h ) s + m h ~ v in( h )1 Y s =1 v out( h )2 Y t =1 ( λ ′ in( h ) s − λ ′′ out( h ) t − m h ∗ ~ (cid:17) . We define the shuffle product SH ~v ⊗ C [ ~ ] SH ~v → SH ~v + ~v by f ( λ ~v ) ⊗ g ( λ ~v ) X σ ∈ Sh( ~v , ~v ) σ (cid:16) f ( λ ′ ~v ) · g ( λ ′′ ~v ) · fac ~v , ~v (cid:17) , (1)where Sh( ~v , ~v ) := Q i ∈ I Sh( v i , v i ), and Sh( v i , v i ) consists of ( v i , v i )-shuffles, i.e., permutations of { , · · · , v i + v i } that preserve the relative order of { , · · · , v i } and { v i + 1 , · · · , v i + v i } .This fac ~v , ~v is slightly different than the one in [YZ14, YZ16] in that here the sign twist in[YZ14] is absorbed into the fac ~v , ~v used here.A special case of the following fact is discussed in [YZ18, Example 3.6]. Due to technical reasonwe also need to consider another version of the shuffle algebra SH ′ , whose underlying vector spaceis a localization of that of SH, denoted by SH ′ = L ~v ∈ N I (SH ′ ~v ) loc to differentiate it from SH. Themultiplication f ( λ ~v ) ⋆ ′ f ( λ ~v ) is defined to be X { σ ∈ Sh ( v ,v ) } σ · (cid:16) f · f · Y i ∈ I v i Y s =1 v i Y t =1 λ ′ is − λ ′′ it + ~ λ ′ is − λ ′′ it Y i,j ∈ I v i Y s =1 v j Y t =1 λ ′ jt − λ ′′ is − a ij ~ λ ′ jt − λ ′′ is + a ij ~ (cid:17) . (2) Proposition 1.2.
There is an algebra homomorphism M ~v ∈ N I SH ~v → M ~v ∈ N I (SH ′ ~v ) loc , given by f ( λ ~v ) f ( λ ~v ) H ~v ( λ ~v ) , where H ~v ( λ ~v ) := Q h ∈ H Q v out( h ) s =1 Q v in( h ) t =1 ( λ in( h ) t − λ out( h ) s + m h ~ ) . In particular, when restricting on SH ~e k , the above homomorphism is an identity, where ~e k is the dimension vector valued at vertex k and zero otherwise. Y. YANG AND G. ZHAO
Here the localization is taken in the obvious sense. As a free polynomial ring is an integraldomain, this map described above is injective.
Proof.
This map is well-defined, since the factor H ~v ( λ ~v ) is invariant under the action of Sh( ~v , ~v ).Define H cross , ~v , ~v := H ~v ~v ( λ ~v ∪ λ ~v ) H ~v ( λ ~v ) H ~v ( λ ~v ) , we have H cross , ~v , ~v = Y h ∈ H v out( h )2 Y s =1 v in( h )1 Y t =1 ( λ ′ in( h ) t − λ ′′ out( h ) s + m h ~ v out( h )1 Y s =1 v in( h )2 Y t =1 ( λ ′′ in( h ) t − λ ′ out( h ) s + m h ~ ~v , ~v by H cross , ~v , ~v , we obtainfac ~v , ~v H cross , ~v , ~v = Y i ∈ I v i Y s =1 v i Y t =1 λ ′ is − λ ′′ it + ~ λ ′ is − λ ′′ it Y h ∈ H v out( h )2 Y s =1 v in( h )1 Y t =1 λ ′ in( h ) t − λ ′′ out( h ) s − m h ∗ ~ λ ′ in( h ) t − λ ′′ out( h ) s + m h ~ = Y i ∈ I v i Y s =1 v i Y t =1 λ ′ is − λ ′′ it + ~ λ ′ is − λ ′′ it Y i,j ∈ I a ij Y p =1 v i Y s =1 v j Y t =1 λ ′ jt − λ ′′ is + ( a ij − p ) ~ λ ′ jt − λ ′′ is + ( a ij + 2 − p ) ~ = Y i ∈ I v i Y s =1 v i Y t =1 λ ′ is − λ ′′ it + ~ λ ′ is − λ ′′ it Y i,j ∈ I v i Y s =1 v j Y t =1 λ ′ jt − λ ′′ is − a ij ~ λ ′ jt − λ ′′ is + a ij ~ . It implies f ( λ ~v ) ⋆ f ( λ ~v ) H ~v ( λ ~v ) = f ( λ ~v ) H ~v ( λ ~v ) ⋆ ′ f ( λ ~v ) H ~v ( λ ~v ) . This completes the proof. (cid:3)
For each k ∈ I , recall that ~e k is the dimension vector valued 1 at vertex k and zero otherwise.That is, ~e k corresponds to the simple root α k of g KM . Definition 1.3.
We define the spherical subalgebra SH sph to be the subalgebra of SH generatedby SH ~e k as k varies in I .In [YZ16, § D (SH sph , ext ) of SH. The constructionis given as follows. Roughly speaking, let SH := Sym( h [ u ]) be the symmetric algebra of h [ u ], whichacts on SH sph by [YZ16, Lemma 1.4]. Let SH ext , sph := SH ⋉ SH sph be the extended shuffle algebra.SH ext , sph is a bialgebra endowed with a non-degenerate bialgebra pairing, whose coproduct is givenin [YZ16, Theorem 2.2] and the pairing in [YZ16, Theorem 3.5]. Let D (SH sph , ext ) be the Drinfelddouble of SH sph , ext , as a vector space one has D (SH sph , ext ) ∼ = SH sph , ext ⊗ (SH sph , ext ) coop . Definethe reduced Drinfeld double D (SH sph , ext ) to be the quotient of D (SH sph , ext ) by identifying SH inSH sph , ext with (SH ) coop in (SH sph , ext ) coop . See [YZ16, § D (SH sph , ext ) ∼ = SH sph ⊗ SH ⊗ SH sph , coop . The pairing ( , ) SH on SH sph , ext . This part follows from the results in [YZ16]. We brieflyrecall the non-degenerate pairing on the extended shuffle algebra SH sph , ext = SH sph ⋊ SH in [YZ16,Section 3]. HE PBW THEOREM FOR THE AFFINE YANGIANS 5
Let SH sph , ext A := SH sph A ⋊ SH A be the ad´ele version of the shuffle algebra defined in detail in[YZ14, § sph , ext A is defined as follows.( · , · ) : SH sph , ext A ⊗ (SH sph , ext A ) coop → C • For f ∈ SH sph , ext A ,~v , and g ∈ SH sph , ext A , ~w , we define ( f, g ) = 0 if ~v = ~w . • For h ∈ SH , and f ∈ SH sph A , we define ( h, f ) = 0. • For H k ( u ) ∈ SH [[ u ]], we define ( H k ( u ) , H k ( w )) = fac( u | w )fac( w | u ) , for any k ∈ I , where H k ( u ) = 1 + ~ P r ≥ h k,r u − r − is the generating series of the standard basis { h k,r = h k ⊗ λ r | k ∈ I, r ∈ N } of SH = Sym( h [ λ ]). Here fac( u | w ) is fac ~e k , ~e k with the variables λ k ′ and λ k ′′ replaced by u and w respectively. • For any i ∈ I and f, g ∈ SH A ,~e i , we follow Drinfeld [D86] and define( f, g ) := Res x = ∞ ( f x · g − x dx ) . In general, for f, g ∈ SH sph , ext A ,~v , the pairing ( f, g ) is defined to be( f, g ) := X x ∈ P Res x (cid:16) f ( x ~v ) · g ( − x ~v ) Q i ∈ I ( v i )! fac( x ~v ) dx (cid:17) , wherefac( x ~v ) := Y i ∈ I Y { s,t | ≤ s = t ≤ v i } λ ′ is − λ ′′ it + ~ λ ′′ it − λ ′ is ·· Y h ∈ H (cid:16) v out( h ) Y s =1 v in( h ) Y t =1 ( λ ′′ in( h ) t − λ ′ out( h ) s + m h ~ v in( h ) Y s =1 v out( h ) Y t =1 ( λ ′′ out( h ) t − λ ′ in( h ) s + m h ∗ ~ (cid:17) . This pairing on SH sph A ⊗ SH sph , coop A described above is a non-degenerate bialgebra pairing, whichrestricts to a non-degenerate one when ~ = 0. In particular, on the set of generators, we havefac( x ~e i ) = 1, and (( λ ( i ) ) r , ( λ ( j ) ) s ) = δ ij Res x = ∞ ( x r · ( − x ) s dx )=( − s δ ij δ r + s, − . Theorem 1.4. [YZ16, Theorem B]
Let Y ~ ( g KM ) be the Yangian endowed with the Drinfeld comul-tiplication. Assume Q has no edge-loops. Then there is a bialgebra epimorphism Ψ : Y ~ ( g KM ) ։ D (SH sph , ext ) . When Q is of finite type, this map is an isomorphism. Composing Ψ with the natural map Y + ~ ( g KM ) → Y ~ ( g KM ), we get the map Ψ : Y + ~ ( g KM ) ։ SH sph , which sends x + k,r to ( λ ( k ) ) r ∈ SH ~e k = C [ ~ ][ λ ( k ) ], where k ∈ I . Finally we recall that from[YZ16, § Y − ~ ( g KM ) → Y ~ ( g KM ) we get a surjective algebrahomomorphism(3) Y ≤ ~ ( g ) → (SH sph , ext ) coop defined by sending x − i,r to ( − λ ( i ) ) r and ξ k,s to − ( − h ( k ) ) s . Y. YANG AND G. ZHAO
The main theorems.
From now on till the end of this paper, we take g KM to be of untwistedsymmetric affine type . The simple roots I are labeled by 0 , , · · · , n with 0 being the extendednode in the affine Dynkin diagram.In the present paper, we prove the following. Theorem A.
Assume the symmetric Cartan matrix A is of nontwisted affine Kac-Moody typeand not of type A (1)1 . The morphism Ψ : Y ~ ( g KM ) → D (SH sph , ext ) is an isomorphism.Theorem A is proved in a similar fashion to [E03] and [Ne15]. In [E03], the PBW theorem forthe quantum Kac-Moody algebras is proved using an isomorphism with the Feigin-Odesskii shufflealgebras. In [Ne15], an isomorphism of the quantum toroidal algebra of gl n and the double shufflealgebra of Feigin-Odesskii is established.As a consequence of Theorem A, we have the following PBW theorem of the affine Yangian Y ~ ( g KM ).Define a filtration on Y ~ ( g KM ) by deg x ± i,r ≤ r and deg h i,r ≤ r for all i ∈ I and r ∈ N . Letgr( Y ~ ( g KM )) be the associated graded algebra. Let t be the universal central extension of g KM [ u ]. Corollary A.
We have the natural isomorphism gr( Y ~ ( g KM )) ∼ = U ( t )[ ~ ].In §
4, we also prove a triangular decomposition of Y ~ ( g KM ).2. The positive part
In this section, we focus on the positive part Y + ~ ( g KM ) of the Yangian, and study the mapΨ : Y + ~ ( g KM ) ։ SH sph .2.1. Preliminaries on the current algebras.
We collect some facts from [E03, § g fin bethe finite dimensional Lie algebra of g KM . Let g fin = n − fin ⊕ h fin ⊕ n fin be the triangular decompositionof g fin . Then, n KM = ( λ C [ λ ] ⊗ ( n − fin + h fin ) + C [ λ ] ⊗ n fin ) [K85, § A presentation.
The set up in [E03, § n KM [ u, u − ]. For our pur-poses, we state the results for the current algebra n KM [ u ].Define t + to be the direct sum n KM [ u ] ⊕ L k> ,l ∈ N C K kδ [ l ], and endow it with the bracket suchthat the elements K kδ [ l ] are central, and[( x ⊗ u l , , ( y ⊗ u m , x ⊗ u l , y ⊗ u m ] , h ¯ x, ¯ y i ( lk ′′ − mk ′′ ) K ( k ′ + k ′′ ) δ [ l + m ]) , where x ¯ x ⊗ λ k ′ , y λ k ′′ by the inclusion n KM ⊂ g fin [ λ, λ − ], and h ¯ x, ¯ y i is an invariant bilinearform on g fin . Definition 2.1.
Let ˜ F + be the Lie algebra with generators x + i,k , i ∈ I, k ∈ N , and relations givenby (Y4) with ~ = 0, and (Y6) . In other words, U ( ˜ F + ) = Y + ~ =0 ( g KM ).It is straightforward to see that there is a Lie algebra morphism j + : ˜ F + → n KM [ u ] , x + i,k x i ⊗ u k . Proposition 2.2. [E03, Proposition 1.6] (1) When A is of affine Kac-Moody type, the kernel j + is equal to the center of ˜ F + , so that ˜ F + is a central extension of n KM [ u ] .(2) We have a unique Lie algebra map j ′ : t + → ˜ F + such that j ′ ( e i ⊗ t k ) = x + i,k , where i = 0 , , · · · , n This map is an isomorphism iff A is not of type A (1)1 . If A is of type A (1)1 , j ′ is surjective, and its kernel is ⊕ k ∈ N C K δ [ k ] . HE PBW THEOREM FOR THE AFFINE YANGIANS 7
As a consequence, when A is of affine Kac-Moody type and not of type A (1)1 , there is an algebrahomomorphism U ( t + ) → U ( n KM [ u ]) extending the identity map on n KM [ u ]. The kernel of this mapis generated by L k> ,l ∈ N K kδ [ l ].2.1.2. The pairing ( , ) t on U ( t + ) . This part follows from the results in [E03].We first recall the pairing defined on a toroidal algebra in [E03, Remark 4.3]. Let tor be theuniversal central extension of g fin [ λ ± , u ± ]. Then, tor = g fin [ λ ± , u ± ] ⊕ Z, where Z = Ω C [ λ ± ,u ± ] /d ( C [ λ ± , u ± ]) = L k,l ∈ Z K kδ [ l ] ⊕ C c , and K kδ [ l ] is the class of k λ k u l − du if k = 0, K [ l ] is the class u l dλλ , c is the class duu . We decompose Z as Z = Z > ⊕ Z ⊕ Z < , where Z > := M k> ,l ∈ Z K kδ [ l ], Z := M l ∈ Z K [ l ] ⊕ C c and Z < := M k< ,l ∈ Z K kδ [ l ].Enlarge tor by ˜ tor := g fin [ u ± , λ ± ] ⊕ Z ⊕ D , where D = D > ⊕ D ⊕ D < = ( M k> ,l ∈ Z D kδ [ l ]) ⊕ ( M l ∈ Z D [ l ] ⊕ C d ) ⊕ ( M k< ,l ∈ Z D kδ [ l ]) . The following pairing on ˜ tor can be found in [E03, Remark 4.3].
Proposition 2.3.
There is a non-degenerate bialgebra pairing ( , ) ˜ tor on ˜ tor = g fin [ λ ± , u ± ] ⊕ Z ⊕ D .Let x, x ′ ∈ g fin , the pairing ( , ) ˜ tor is the following. ( x ⊗ λ a u b , x ′ ⊗ λ a ′ u b ′ ) ˜ tor = h x, x ′ i g fin δ a + a ′ , δ b − b ′ , , ( K kδ [ l ] , D k ′ δ [ l ′ ]) ˜ tor = δ k + k ′ , δ l − l ′ , , ( d, c ) ˜ tor = 1 . and all other pairings of elements x ⊗ λ a u b , K kδ [ l ] , D kδ [ l ] , c, d are zero. Note that we follow the same convention as in [YZ16, § t + is isomorphic to the Lie subalgebra of ˜ tor [E03, pg. 29] n KM [ u ] ⊕ M k> ,l ∈ N C K kδ [ l ] = ( λ C [ λ ] ⊗ ( n − fin + h fin ) + C [ λ ] ⊗ n fin )[ u ] ⊕ M k> ,l ∈ N C K kδ [ l ] . Let t − ⊂ ˜ tor be the subalgebra n − KM [ u ] ⊕ L k< ,l ∈ N C D kδ [ l ] = ( λ − C [ λ − ] ⊗ ( n +fin + h fin ) + C [ λ − ] ⊗ n fin )[ u ] ⊕ L k< ,l ∈ N C D kδ [ l ] . Restricting the pairing in Proposition 2.3, we get a non-degenerate pairing t + × t − → C . Wenow describe this paring on the set of generators. We have, for i, j = 0 , , · · · , n ,(4) ( x + i,r , x − j,s ) t = δ ij δ r − s, , ( K kδ [ l ] , D k ′ δ [ l ′ ]) = δ k + k ′ , δ l − l ′ , . Indeed, when i = j = 0, x +0 ,r corresponds to E − θ ⊗ λu r , and x − ,s corresponds to E θ ⊗ λ − u s , where θ is the highest root of g fin , and E θ ∈ g fin ,θ [K85, § x +0 ,r , x − ,s ) t = δ ij δ r − s, .This pairing identifies t + with the dual of t − endowed with the opposite coproduct. Note thathowever, t + is isomorphic to t − as algebras via the following map(5) E i, − k − F i,k for all i = 0 , . . . , n and k ∈ Z . Using the presentation of t + given in Proposition 2.2, one easily verifythat the relations hold in t − by comparing with [GTL10, § H i,r
7→ − H i, − r extends the above isomorphism to the Cartan subalgebra. Y. YANG AND G. ZHAO
The statement.
Motivated by the filtration on Y + ~ ( g KM ), we define a filtration F = { F r } r ∈ N on SH sph , such that ( λ ( k ) ) r is in the filtered piece F r . As various filtrations on SH sph will beconsidered, we denote this filtration by F = { F r } r ∈ N and the corresponding associated gradedalgebra by gr F (SH sph ). That is, the generator ( λ ( k ) ) r is in the filtered piece F r , and ~ ∈ F . Recallthat the algebra homomorphism in Theorem 1.4 is a filtered map, hence gives rise to a surjectivemorphism on the associated graded algebrasgr(Ψ) : gr( Y + ~ ( g KM )) ։ gr F (SH sph ) . Using the isomorphism U ( t + ) ∼ = U ( ˜ F + ) and the presentation of the latter in Definition 2.1, we havean algebra epimorphism π : U ( t + )[ ~ ] ∼ = U ( ˜ F + )[ ~ ] ։ gr( Y + ~ ( g KM )) , e i ⊗ t k x + i,k , i = 0 , , · · · , n . Theorem 2.4.
The composition U ( t + )[ ~ ] π / / / / gr( Y + ~ ( g KM )) gr(Ψ) / / / / gr F (SH sph ) is an isomorphism. We will prove Theorem 2.4 in §
3. By Theorem 1.4, Ψ is an epimorphism. It follows fromthis theorem that gr(Ψ) is an isomorphism, and consequently that Ψ : Y + ~ ( g KM ) ∼ = SH sph is anisomorphism. 3. Proof of Theorem 2.4
We prove Theorem 2.4 in the following steps.3.1.
Step 1.
Note that Y + ~ ( g KM ) | ~ =0 ∼ = U ( t + ) by Proposition 2.2. In this step, we show thefollowing. Proposition 3.1.
The map Ψ | ~ =0 : U ( t + ) ։ SH sph | ~ =0 is an algebra isomorphism. The map Ψ | ~ =0 is compatible with the coproducts:(6) Ψ | ~ =0 (∆( x )) = ∆(Ψ | ~ =0 ( x )) , for x ∈ t + . In [YZ16, Section 2], we defined a coproduct ∆ : SH ext , sph → SH ext , sph ˆ ⊗ SH ext , sph on the extendedshuffle algebra. In particular, by [YZ16, (8)],∆(( λ ( i ) ) r ) = H i ( λ ( i ) ) ⊗ ( λ ( i ) ) r + ( λ ( i ) ) r ⊗ . When specializing ~ = 0, we have on SH sph ~ =0 , ∆(( λ ( i ) ) r ) = 1 ⊗ ( λ ( i ) ) r + ( λ ( i ) ) r ⊗
1. As the coproducton U ( t + ) is given by ∆( x + i,r ) = 1 ⊗ x + i,r + x + i,r ⊗
1, we have, on the level of generators,Ψ | ~ =0 (∆( x + i,r )) = ∆(( λ ( i ) ) r ) = ∆(Ψ | ~ =0 ( x + i,r )) . The claim (6) now follows from the fact that ∆ and Ψ | ~ =0 are algebra homomorphisms.We prove Proposition 3.1 using the non-degenerate bialgebra pairings on U ( t + ) and SH sph | ~ =0 described below. The non-degenerate bialgebra pairings on U ( t + ) and SH sph | ~ =0 are denoted by( , ) t and ( , ) SH ~ =0 respectively. The paring ( , ) SH ~ =0 is recollected in § , ) t , and the compatibility of the map Ψ | ~ =0 with the parings. HE PBW THEOREM FOR THE AFFINE YANGIANS 9
We have the following commutative diagram of algebras U ( t + ) coop Ψ coop | ~ =0 / / SH sph , coop ~ =0 U ( t − ) coop ∼ = O O Y − ~ =0 ( g ) ∼ = o o O O where the left vertical map is from (4); the right vertical map is from [YZ14, § coop sends x + j,s to ( − s ( λ ( j ) ) s . The commutativity of this diagram is clear by keeping track ofthe generators of Y − ~ ( g ).Identifying t − with t + using the map (5) x − i,k ↔ x + i, − k − , we get the pairing ( , ) t on t + using theparing described in § x + i,r , x + j,s ) t = ( x + i,r , x − j, − − s ) t + × t − = δ ij δ r + s +1 , . Therefore,(7) ( x + i,r , x + j,s ) t = (Ψ | ~ =0 ( x + i,r ) , Ψ coop | ~ =0 ( x + j,s )) SH ~ =0 = (( λ ( i ) ) r , ( − s ( λ ( j ) ) s ) SH ~ =0 = δ ij δ r + s, − . Moreover, they are bialgebra pairings, i.e.,(8) ( a ⋆ b, c ) = ( a ⊗ b, ∆( c )) , for a, b, c ∈ A ,where A = t + or SH ~ =0 . Therefore, it follows from (6), (7), and (8) that the map Ψ | ~ =0 is compatiblewith the pairings on the entire U ( t + ) and SH sph | ~ =0 .The injectivity of Ψ | ~ =0 follows from the nondegeneracy of the two pairings when extended tothe ad´ele version and the fact that the map Ψ | ~ =0 : U ( t + ) ։ SH sph | ~ =0 preserves the pairing.Indeed, for any x ∈ U ( t + ) such that Ψ | ~ =0 ( x ) = 0. We have, for any y ∈ U ( t + ),( x, y ) t = (Ψ | ~ =0 ( x ) , Ψ coop | ~ =0 ( y )) SH ~ =0 = 0 . It implies x = 0 by non-degeneracy. Therefore, Ψ | ~ =0 is injective. This concludes Proposition 3.1.3.2. Step 2.
We define another filtration on SH sph by the degrees of ~ . To be more precisely, weassign deg( ~ ) = −
1, deg(( λ ( i ) ) r ) = 0, and extend it to a filtration on SH sph . Let gr ~ (SH) be theassociated graded of SH using this ~ -filtration.In this subsection, we show the following. Proposition 3.2.
We have an isomorphism of algebras gr ~ (SH) ∼ = SH | ~ =0 [ ~ ] . Proof.
Let F ′ = { F ′ r } r ∈ Z ≤ be the the filtration induced by ~ , defined as deg(( λ ( k ) ) r ) = 0, anddeg( ~ ) = −
1. Then, F ′ r consists of those elements in SH whose ~ n term satisfies n ≥ − r . Therefore, F ′ r /F ′ r − consists of those elements in SH whose ~ n term satisfies n = − r . As a consequence, wehave the isomorphism F ′ r /F ′ r − ∼ = (SH | ~ =0 ) ~ − r . We note that this isomorphism, which relies on thefact that SH is flat as a C [ ~ ]-module, is the key to the proof of the PBW theorem in the presentpaper.We check that this isomorphism preserves the algebra structure. For any x, y ∈ gr ~ (SH), withoutloss of generality, we can assume the leading terms of x, y have degree 0. Otherwise, we can shift x, y by ~ n , for some n ∈ N . The multiplication x ⋆ y in gr ~ (SH) is by throwing away the ~ -termsin x ⋆ y ∈ SH. This is the same as setting ~ = 0. Therefore, it is the same as the multiplication inSH | ~ =0 [ ~ ]. This finishes the proof. (cid:3) Step 3.
We now show the following statement.
Proposition 3.3.
We have an isomorphism of algebras gr F (SH sph ) ∼ = gr ~ (SH sph ) . Note that this isomorphism of algebras does not preserve the gradings.
Proof.
Now we consider a third grading on SH sph defined as follows. Let SH ′ be the localizedshuffle algebra considered in § ~ and λ ( k ) t . In particular, deg(( λ ( k ) ) r ) = r , and deg( ~ ) = 1. Note that the multiplicationformula (2) makes SH ′ into a graded algebra with the degree- m piece denoted by G ′ m for m ∈ Z .Using the embedding from Proposition 1.2, we define the grading on SH sph via G m = G ′ m ∩ SH sph .We therefore have SH sph = L m ∈ N G m .Recall the filtration F on SH sph defined by deg(( λ ( k ) ) r ) ≤ r , and deg( ~ ) = 0. It induces afiltration on G m for each m ∈ Z ( F m ∩ G m ) ⊇ ( F m − ∩ G m ) ⊇ · · · ⊇ ( F ∩ G m ) . Similarly, let F ′ = { F ′ r } r ∈ Z ≤ be the the filtration induced by ~ , defined as deg(( λ ( k ) ) r ) = 0, anddeg( ~ ) = −
1. The filtration F ′ on SH sph induces a filtration on G m :( F ′ ∩ G m ) ⊇ ( F ′− ∩ G m ) ⊇ · · · ⊇ ( F ′− m ∩ G m ) . Both F and F ′ are compatible with G in the sense that F m = ⊕ i ( G i ∩ F m ) and F ′ m = ⊕ i ( G i ∩ F ′ m ).We have the following observation, for 0 ≤ r ≤ m , F r ∩ G m = F ′ r − m ∩ G m . Indeed, we have x ∈ F r ∩ G m ⇐⇒ x is of the form x = X { s | s ≤ r } x s ~ m − s , where deg( x s ) = s . ⇐⇒ x ∈ F ′ r − m ∩ G m , where the last implication follows from the fact that the power of ~ in x showing up above are allof the form m − s ≥ m − r . This implies the isomorphism. m M r =0 ( F r ∩ G m ) / ( F r − ∩ G m ) ∼ = M r = − m ( F ′ r ∩ G m ) / ( F ′ r +1 ∩ G m ) . Therefore, we have gr F (SH sph ) = M m ≥ m M r =0 ( F r ∩ G m ) / ( F r − ∩ G m ) ∼ = M m ≥ M r = − m ( F ′ r ∩ G m ) / ( F ′ r − ∩ G m )= gr ~ (SH sph ) . The algebra structures on gr F (SH sph ) and gr ~ (SH sph ) are both induced from the product on SH sph .Therefore, we conclude the claim. (cid:3) HE PBW THEOREM FOR THE AFFINE YANGIANS 11
Conclusion.
We put Propositions 3.1, 3.2, 3.3 together. The morphism gr(Ψ) ◦ π in Theorem2.4 is the following composition U ( t + )[ ~ ] ∼ = SH sph | ~ =0 [ ~ ] ∼ = gr ~ (SH sph ) ∼ = gr F (SH sph ) . Therefore, gr(Ψ) ◦ π is an isomorphism. This completes the proof of Theorem 2.4.4. Triangular decomposition
Recall we have the natural map Ψ : Y ~ ( g KM ) → D (SH sph , ext ) ∼ = SH sph ⊗ SH ⊗ SH sph , coop fromTheorem 1.4, where the last isomorphism is only of vector spaces. Composing with the naturalmap Y + ~ ( g KM ) → Y ~ ( g KM ), we get the map Y + ~ ( g KM ) → SH sph ⊗ SH ⊗ SH sph , coop , which is injectivethanks to Theorem 2.4. When restricting on Y , the map Ψ : Y ~ ( g KM ) → SH is an isomorphismby the definitions of both sides.In this section, we prove the following theorem. Theorem 4.1.
The map
Ψ : Y ~ ( g KM ) → SH sph ⊗ SH ⊗ SH sph , coop is an isomorphism of vectorspaces. In particular, the natural map Y + ~ ( g KM ) ⊗ Y ⊗ Y − ~ ( g KM ) → Y ~ ( g KM ) induced by multipli-cations is an isomorphism of vector spaces. This is the triangular decomposition of the Yangian.
Proof.
It follows from the commutation relation of Y + ~ ( g KM ) and Y (see, e.g., [GTL10, Lemma 2.9])that the natural map Y + ~ ( g KM ) ⊗ Y → Y ≥ ~ ( g KM ) induced by multiplication is surjective. Bysymmetry, Y − ~ ( g KM ) ⊗ Y → Y ≤ ~ ( g KM ) is also surjective. Hence, the commutation relation of Y + ~ ( g KM ) and Y − ~ ( g KM ) (Y5) implies that the natural map Y + ~ ( g KM ) ⊗ Y ⊗ Y − ~ ( g KM ) → Y ~ ( g KM )is surjective.Compositing this surjective map with the algebra epimorphism Ψ : Y ~ ( g KM ) → D (SH sph , ext )from Theorem 1.4, we get Y + ~ ( g KM ) ⊗ Y ⊗ Y − ~ ( g KM ) → Y ~ ( g KM ) → D (SH sph , ext ) ∼ = SH sph ⊗ SH ⊗ SH sph , coop . Theorem 2.4 yields that the composition is an isomorphism of vector spaces.Therefore, Y + ~ ( g KM ) ⊗ Y ⊗ Y − ~ ( g KM ) ∼ = Y ~ ( g KM ) and Y ~ ( g KM ) ∼ = D (SH sph , ext ). This completesthe proof. (cid:3) References [D86] V. G. Drinfeld,
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